Quadratic Equations
Quadratic Equations
Quadratic equations are fundamental in algebra and are widely used in various fields of science, engineering, and mathematics. They can be represented in the form:
a${x}^{2}$ + bx + c = 0
Where:
 x is the variable you want to solve for.
 a, b, and c are constants, with a ≠ 0.
1. Solving Quadratic Equations:
a. Quadratic Formula:
 The quadratic formula is a universal method to solve any quadratic equation. It is given by:
x = (b ± √(${b}^{2}$ 4ac)) / (2a)

The "±" symbol means that there are usually two solutions: one with the plus sign and one with the minus sign. These solutions are often referred to as the "roots" of the equation.

Use this formula to find the solutions (roots) of the quadratic equation.
b. Factoring:  If the quadratic equation can be factored into two binomials, it is easy to solve.
Example: If you have ${x}^{2}$  5x + 6 = 0, it can be factored as (x  2)(x  3) = 0.
c. Completing the Square:  Convert the quadratic equation into the form ${(xp)}^{2}$ = q and then solve for x.
Example: For ${x}^{2}$  6x  8 = 0, complete the square to get ${(x3)}^{2}$ = 17.
2. Discriminant:

The discriminant, D, is a value calculated from the coefficients of a quadratic equation: D = ${b}^{2}$  4ac. It provides information about the nature of the roots.

If D > 0, the equation has two distinct real roots.

If D = 0, the equation has one real root (a repeated root).

If D < 0, the equation has two complex (nonreal) roots.
3. Quadratic Equations in Real Life:
 Quadratic equations often represent physical phenomena, such as the motion of projectiles, the shape of parabolic mirrors, or financial modeling. They are also used in optimization problems.
4. Graph of Quadratic Equations:

The graph of a quadratic equation is a parabola.

The direction (upward or downward) and width of the parabola depend on the sign and magnitude of the coefficient a.

The vertex form of a quadratic equation is y = a${(xh)}^{2}$ + k, where (h, k) is the vertex of the parabola.
5. Quadratic Inequalities:

Quadratic inequalities involve quadratic expressions and can have multiple solutions.

Solve them by determining the regions of the number line where the quadratic expression is greater than, less than, or equal to a specific value.